Plane waves are waves with phase fronts that are planes in three-dimensional space and where the energy propagates in a linear direction [Angelsen, 2000].
Plane waves do not exist in practical life, but it can serve as an approximation for a limited region of space when the transversal dimensions can be assumed to be infinite as an example.
An ultrasound transducer generates plane, longitudinal pressure waves into the bulk of the biological material. Longitudinal waves means that the particle motion is along the propagation direction of the wave. The biological mate-rial can be viewed as an isotropic and homogeneous elastic matemate-rial. Isotropic means that the properties are independent of the direction, while homogeneous means that the material properties are constant in space [Angelsen, 2000].
The ultrasound waves can be reflected, absorbed, refracted and focused.
They are not absorbed much by water and other tissues and therefore, the waves can transmit energy into cells. The waves causes cyclic compressions and expansions. In Figure 2.2, we see two fictitious planes, I1 and I2, in the elastic material which are normal to the direction of the wave propagation. The wave motion will cause the planes to vibrate. The displacement of the plane, ψ(z, t), from its equilibrium positionxare shown in the figure. At any time, the vibration velocity and acceleration of the plane can be consequently written as
u(z, t) =∂ψ(z, t)
∂t , (2.1)
a(z, t) = ∂u(z, t)
∂t = ∂2ψ(z, t)
∂t2 . (2.2)
This is the Lagrangian description which, in contrast to the Euler descrip-tion, is based on relating the displacement of a particle in space to an equi-librium position. The equiequi-librium position is the position where no forces are acting upon the particle.
Figure 2.2: Volume change of a thin element of a material caused by the wave motion [Angelsen, 2000]
Pressure, p(z, t), is generated on the plane when the variations of ψ along the propagation direction causes a compression or an expansion of the medium.
The linearized volume compression of the element is defined as δV = [ψ(z, t) + ∆z∂ψ(z, t)
∂z −ψ(z, t)]A. (2.3) Then we can get the relative volume compression as
δV
∆V =∂ψ
∂z, (2.4)
where the unstrained volume is ∆V = ∆zA. The acoustic pressure on the plane can be written as
p≈ −1 κ
∂ψ
∂z, (2.5)
whereκis the compressibility of the fluid. Differentiating with respect to time, Eq. 2.27 can be written as
∂p(z, t)
∂t =−1 κ
∂u
∂z. (2.6)
The negative pressure gradient gives the net force of the volume element between the two planesI1andI2, written as
D’Alembert gives the solution of the one-dimensional wave equation as u(z, t) =U+(z−ct) +U−(z+ct), (2.9) p(z, t) =P+(z−ct) +P−(z+ct), (2.10) where the arbritary functionsU+,U−,P+andP−represents waves propagating in the positive and the negative z-direction shown in Figure 2.3.
Figure 2.3: Positive and negative propagating waves [Angelsen, 2000]
Between two semi-infinite materials with characteristic impedanceI1andI2, we are assuming an interface atz= 0. As shown in Figure 2.4 a wave in the left material propagates from left to the right. When it hits the interface, the wave will be partially reflected and partially transmitted. The reflection coefficient, R01, can be calculated as
R01=P0−
P0+
=Z1−Z0 Z0+Z1
=−U0−
U0+
, (2.11)
and the transmission coefficient,T01, as T01=P1+
P0+
= 2Z1 z0+Z1
. (2.12)
Figure 2.4: Reflections of a plane wave at the boundary between two materials [Angelsen, 2000]
In the left material the pressure and the particle velocity atz= 0 will be
P0=P0++P0−, (2.13)
The pressure and the particle velocity in the right material will be at the inter-face, respectively
P1=P1+, (2.15)
U1= P1+
Z1
. (2.16)
The wave function can also be written as
p(z, t) =Z0[U+(z−ct)−U−(z+ct)], (2.17) where the characteristic mechanical/acoustic impedance, Z0 kg/(m2s) of the material is defined as
Z= 1
κc =ρc. (2.18)
Then we can calculate the reflection coefficient,R01, as R01=P0−
P0+ =Z1−Z0
Z0+Z1 = U0−
U0+, (2.19)
and the transmission coefficient, T01, as T01=P1+
P0+ = 2Z1
Z0+Z1. (2.20)
WhenZ ⇒ ∞andZ⇒0 we get total reflection withR01= 1 andR01=−1 respectively. When the wave propagates from a material with high impedance
to a material with low impedance, the particle velocity is increased and the pres-sure is reduced. The opposite is happening when the wave is travelling from a material with low impedance to a material with high impedance.
In Figure 2.5 we have a plate submersed in two semi-infinite materials with a wave coming from the left infinity in Material 1. The wave is partially reflected and partially transmitted at the interface between material 1 and material 0.
Again, the wave is partially reflected and transmitted at the interface to material 2.
Figure 2.5: Multiple reflections of an incoming wavefront [Angelsen, 2000]
The reflection and transmission coefficients between material 1 and material 0 can be written as
R10=Z0−Z1
Z1+Z0, (2.21)
T10= 2Z0
Z1+Z0
. (2.22)
Likewise, the reflection and the transmission coefficients from material 0 to material 2 are defined as
R02=Z2−Z0 Z0+Z2
, (2.23)
T02= 2Z2 Z0+Z2
. (2.24)
We will have partial reflections back and forth in material 0 as the figure shows. Every time the wave hits the left interface it will generate a transmitted partial wave that propagates leftwards in material 1, and every time the wave hits the right interface it will generate a transmitted partial wave that propa-gates rightwards in material 2.
2.2.1 Energy in acoustic waves
There are two fundamental energy forms in the acoustic wave motion, kinetic energy of the particle velocity and potential energy given by the volume com-pression. With loss of energy from the acoustic wave, energy will be converted to other energy forms like heat and chemical energy. This energy conversion is termed acoustic energy absorption.
To analyze the energy transfer in the wave motion we are considering the volume element in Figure 2.2. There is a change of energy during a short time interval dt given by the product of the displacement of the faces and the force on the faces. The work done on the volume element by the external medium is
dW1=pAudt. (2.25)
Through the right side, the work done is dW2=−[p+∂p
∂z∆z]A[u+∂u
∂z∆z]dt. (2.26)
If the external energy from the work of pressure on the faces is added to the volume element, we get
dW = dW1+ dW2=−∆zA∂p
∂zudt−∆zAp∂u
∂zdt=−∆A∂(pu)
∂z dt. (2.27) This energy is used to build up acoustic wave energy, dE, in the volume element, but some of the energy is also converted to heat, dQ. The energy equation can be expressed as
dW = dE+ dQ. (2.28)
The conservation of energy can be the exchange of energy through either heat conduction or viscous friction. The conversion can also go through relaxation type processes to irregainable forms of energy within the volume element. The conversation to nonregainable heat can be expressed as
dQ=q∆Vdt, (2.29)
whereq(z, t) is the heat current. Inserting Equation 2.27 and 2.28 into 2.29 we get the continuity equation for acoustic energy
−q0=∂e
∂t +∂(pu)
∂z . (2.30)
The acoustic energy density,e=E/∆, is the acoustic energy pr. unit volume.
The power transfer per unit area across an interface normal to the wave direc-tion,p·u, is often denoted as the radiation intensity,I, which is a vector with direction of the particle velocity
I(z, t) =p(z, t)u(z, t)e. (2.31) The acoustic energy density can be written as
e(z, t) = 1
2ρu2(z, t) +1
2κp2(z, t), (2.32)
where the first term in the equation represents the kinetic energy density and the second term represents the potential energy density [Angelsen, 2000]. Dif-ferentiating twice with respect toz, we get
∂2e
∂z2 =−ρκ∂2(pu)
∂z∂t =ρκ∂2e
∂t2 +ρκ∂2q
∂t2. (2.33)
As this equation describes, energy density propagates according to Equation 2.35, which is the wave equation
c= 1